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Theoretical and Methodological Aspects of Music Teacher’S Professional Training O Editorial Board Editor in Chief Mark Zilberman, MSc, Shiny World Corporation, Toronto, Canada Scientific Editorial Board Viktor Andrushhenko, PhD, Professor, Academician of the Academy of Pedagogical Sciences of Ukraine, President of the Association of Rectors of pedagogical universities in Europe John Hodge, MSc, retired, USA Petr Makuhin, PhD, Associate Professor, Philosophy and Social Communications faculty of Omsk State Technical University, Russia Miroslav Pardy, PhD, Associate Professor, Department of Physical Electronics, Masaryk University, Brno, Czech Republic Lyudmila Pet'ko, Executive Editor, PhD, Associate Professor, National Pedagogical Dragomanov University, Kiev, Ukraine Volume 10, Number 2 Publisher : Shiny World Corp. Address : 9200 Dufferin Street P.O. Box 20097 Concord, Ontario L4K 0C0 Canada E-mail : [email protected] Web Site : www.IntellectualArchive.com Series : Journal Frequency : Every 3 months Month : April - June 2021 ISSN : 1929-4700 DOI : 10.32370/IA_2021_06 Trademark © 2021 Shiny World Corp. All Rights Reserved. No reproduction allowed without permission. Copyright and moral rights of all articles belong to the individual authors. Physics M. Pardy The Uniformly Accelerated String and the Bell Spaceship Paradox ……………….... 1 M. Pardy The Vibration of the String with the Interstitial Massive Point …..…..…….…………. 7 History D. Nefyodov, Modern Institutions of General Secondary Education and Institutions of Higher S. Zaskaleta Education …………………………………………………………………………………… 11 Formation and Development of Spiritual And Cultural Centers of the Town of V. Drobnyj Kovalivka on the XIX and Early XX Centuries ……………………………………….…. 16 Psychology V. Semychenko, O. Oleksyuk, Tolerance to Uncertainty in Adolescence as a Psychological Problem ……...……… 29 K. Artyushina Economics I. Partyka System Model of Professional Development Process ………………………………… 37 Law A. Kofanov, N. Pavlovska, O. Romanenko, Ukrainian Economy Sphere and Government Administration Abuse ………………. 43 M. Kulyk, Y. Tereshchenko L. Kotlyarenko, N. Pavlovska, Actual Issues of Forensic Activities in Ukraine and the Possibility of Implementing E. Svoboda, International Standards of the Industry in the Ukrainian Legislation ………….……... 49 A. Symchuk, S. Kharchenko Art Vasyl Avramenko's Creative Heritage in Context Folk Choreography of the XXI O. Bigus Century …………………………………………………………………………...………… 55 continued (continued) I. Ivashchenko Problems of Metaphorization in Modern Directing Theater …………………………… 65 B. Svarnyk Establishment of Eastern European Theater Pantomyms ………….…………….…... 75 Image as a Philosophical Artifact: J. Derrida's Concept in the Context of Modern V. Solomatova Visual Culture ……………………………………………………………………………… 86 Culturology K. Golub Ethnocultural Self-Identification as an Object of Cultural Research …………………. 95 E. Sidorovskaya Cultural Language Activities in the Practice of International Business Etiquette …… 104 Education O. Matviienko, Diagnosis of the Levels of Social Competence Among Elementary School Pupils ... 114 D. Hubarieva Results of an Experimental Test of the Establishment of Readiness of Future A. Yurkov Officers-Psychologists to Serve in the Armed Forces of Ukraine ………..………….. 130 The Dual Education System as a Key Element for Future Railway Experts at the O. Kovalenko Beginning of the 21st Century ……………………………………………………………. 138 Research of Artistic and Creative Activity as a Means of Forming the Humanistic Qian Kai Culture of Students ………………………………………………………………………... 146 Theoretical and Methodological Aspects of Music Teacher’s Professional Training O. Kondratiuk in Higher Education Institutions …………………………………………………………. 153 Technology A. Berhulov Planetary Magnetic Engine ……………………………………………………..……..… 165 Manuscript Guidelines …………………………………………………………………….. 173 DOI: 10.32370/IA_2021_06_1 THE UNIFORMLY ACCELERATED STRING AND THE BELL SPACESHIP PARADOX Miroslav Pardy Department of Physical Electronics Masaryk University Kotl´aˇrsk´a2, 611 37 Brno, Czech Republic e-mail:[email protected] June 1, 2021 Abstract We consider the string with the length l, the left end and the right end of which is non-relativistically and then relativistically accelerated by the constant acceleration a. We calculate the motion of the string with no intercalation of the Fitzgerald contraction of the string. We consider also the Bell spaceship paradox. The Bell paradox and our problem is in the relation with the Lorentz contraction in the Cherenkov effect (Pardy, 1997) realized by the carbon dumbbell moving in the LHC or ILC (Pardy, 2008). The Lorentz contraction and Langevin twin paradox (Pardy, 1969) is interpreted as the Fock measurement procedure (Fock, 1964;). 1 Introduction We will consider the string with the length l, the left end of which is accelerated by the constant acceleration a and the right end is accelerated also by the constant acceleration a. We will calculate the motion of such accelerated string. The differential equation of motion of string element can be derived as it follows. We suppose that the force acting on the element dl of the string is given by the law (Koshlyakov, et al., 1962): @u! T (x; t) = ES ; (1) @x where E is the modulus of elasticity, S is the cross section of the string. We easily derive that 1 IntellectualArchive Vol. 10, No. 2, April - June 2021 1 @u! @u! T (x + dx) − T (x) = ES (x + dx) − ES (x) ≈ ESu dx: (2) @x @x xx The mass dm of the element dl is %Sdx, where % is the mass density of the string matter and the dynamical equilibrium gives %Sdxutt = ESuxxdx: (3) Or, after minimal modification we get 1 E !1=2 u − u = 0; c = : (4) c2 tt xx % The last procedure was performed evidently in order to get the wave equation. 2 The non-relativistic acceleration of the string Our problem is described by the wave equation (Koshlyakov, et al., 1962) 2 utt = c uxx + g(x; t); (5) where g(x; t) = p(x; t)=%, p(x; t) being a force and the boundary conditions are 1 u(x = 0) = κ (t) = at2; (6) 1 2 1 u(x = l) = κ (t) = at2 + l = κ (t) + l: (7) 2 2 1 The initial conditions are u(t = 0) = f(x); ut(t = 0) = F (x): (8) The problem cannot be solved by the standard Fourier method because the bound- ary conditions (6)-(7) are not homogeneous. So, we introduce the auxiliary function (Koshlyakov et al., 1962) x w(x; t) = κ (t) + [κ (t) − κ (t)] (9) 1 2 1 l with the boundary conditions w(x = 0) = κ1(t); w(x = l) = κ1(t) + l (10) and we take the solution in the form: u(x; t) = v(x; t) + w(x; t): (11) with the boundary conditions v(x = 0) = 0; v(x = l) = 0 (12) and with the initial conditions v(t = 0) = f1(x); vt(t = 0) = F2(x): (13) 2 IntellectualArchive Vol. 10, No. 2, April - June 2021 2 After insertion of u = v + w into ew (5), we get the following equation for v and w: 2 2 vtt = c vxx + g(x; t) + c wxx − utt: (14) Then, if we use the definition of w by eq. (9), we get equation for v in the form: 2 vtt = c vxx + g1(x; t); (15) where x g (x; t) = g(x; t) − κ00(t) − [κ00(t) − κ00(t)] (16) 1 1 2 1 l So, we see, that the last algebraic procedures lead to new system of equations. Namely: 2 vtt = c vxx + g1(x; t) (17) with v(x = 0) = 0; v(x = l) = 0 (18) and v(t = 0) = f1(x); vt(t = 0) = F2(x) (19) It is easy to show that g1(x; t) = g − a and the system of equation to be solved is as follows: 2 vtt = c vxx + g − a (20) with v(x = 0) = 0; v(x = l) = 0 (21) and v(t = 0) = f1(x) = 0; vt(t = 0) = F1(x) = 0 (22) The solution of the system is well known (Koshlyakov et al., 1962) and so we write the final form: 1 ! X kπx v(x; t) = Tk sin ; (23) k=1 l where 2 Z t Z l kπξ ! Tk(t) = dτ G(ξ; τ) sin !k(t − τ) sin dξ; (24) l!k 0 0 l where kπc ! = ; G(ξ; τ) = g − a: (25) k l 1 2 So, u = v + w = v + 2 at + x. 3 IntellectualArchive Vol. 10, No. 2, April - June 2021 3 3 The relativistic acceleration of the string Let us consider, at first, the relativistic uniformly accelerated motion of a particle, i.e. the rectilinear motion for which the acceleration w in the proper reference frame (at each instant of time) remains constant. In the reference frame where the particle velocity is v = 0, the components of the four- acceleration is a0 = (0; a=c2; 0; 0), where a is the ordinary three-dimensional acceleration directed along the x axis and c is here the velocity of light. The relativistically invariant condition for uniform acceleration is the constancy of the four-scalar which coincides with a2 in the proper reference frame (Landau, et al., 1987): a2 − = aia ≡ const (26) c4 i In the ”fixed” frame, with respect to which the motion is observed, writing out the i expression for a ai gives the equation d v q = a; (27) dt v2 1 − c2 or, v = at + const; (28) q v2 1 − c2 Setting v = 0 for t = 0, we find that const = 0, so that vt v = = at + const; (29) q a2t2 1 − c2 Integrating once more and setting x = 0 for t = 0, we find: 0s 1 c2 a2t2 x = @ 1 + − 1A (30) a c2 For at c, these formulas go over the classical expressions v = at; x = at2=2. For at ! 1, the velocity tends toward the constant value c. The proper time of a uniformly accelerated particle is given by the integral s Z t v2 c at 1 − = sinh−1 (31) 0; c2 a c As t ! 1 it increases much more slowly than t, according to the law c=a log(2at=c) (Landau, et al., 1987). The solution is the same as in case of the non-relativistic motion, only with the replacing eqs. (6-7) by the equations with the relativistic motion, or with the boundary conditions 0s 1 c2 a2t2 u(x = 0) = κ1(t) = @ 1 + − 1A (32) a c2 0s 1 c2 a2t2 u(x = l) = κ2(t) = @ 1 + − 1A + l = κ1(t) + l (33) a c2 4 IntellectualArchive Vol.
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